The letters of the word ABACUS have been arranged in the shape of a
triangle. How many different ways can you find to read the word
ABACUS from this triangular pattern?

Warmsnug Double Glazing

Stage: 3 Challenge Level:

We received many solutions in which people used pairs of similar windows to find the prices. One possibility is to find a pair with the same area, but different frame lengths. Harry & Roxana from Thorpe House Langley Preparatory School did this:

We looked at K (Area = 12, Frame = 17) and I (Area = 12, Frame = 14) and we used the £60 price difference to find out the cost of the frame (£20 per unit) and the cost of the glass (£10 for each 1 by 1 pane).

Millie and Kate's method for finding the costs of the frame and the glass is very neat:

K had 3 cm of extra frame and was £60 more so we divided it by 3 to find that each centimetre of frame was £20.

Similarly, Jake from Colyton Grammar School started by finding the price of each unit square of glass:

J (Area = 4, Frame = 8) and H (Area = 3, Frame = 8) each have the same length of frame, but J has one square of glass more. J costs £10 more than H so that means that a 1 by 1 pane of glass costs £10.

Once these prices have been found, the correct price of each shape can be found and compared to its price tag. But what would happen if one of K, I, J or H had the wrong price tag? Would this method work? Luckily, many students checked all the price tags and found that the only incorrect one is E.

E has 18 frame squares and 12 glass panes, so it should cost £360 + £120, which equals £480. The price marked is £550, so window E is wrong.

Rhea from Loughborough High School used a very systematic approach to make sure she found the window that was priced incorrectly. She used an algebraic method with simultaneuous equations:

I used X to represent the price of the frame per square and Y to represent the price of the glass per square. For each window I wrote an equation using X and Y:

I then looked for equations which had equal X or Y figures. I used these
equations to explore some simultaneous equations:

F. 12X + 9Y = 330
C. 12X + 8Y = 320
F - C:
Y = 10

J. 8X + 4Y = 200
H. 8X + 3Y = 190
J - H:
Y = 10

N. 20X + 24Y = 640
D. 20X + 16Y = 560
N - D:
8Y = 80
Y = 10

B. 16X + 15Y = 470
O. 16X + 12Y = 440
B - O:
3Y = 30
Y = 10

As all the answers to the simultaneous equations which I investigated are Y = 10, and there is only one incorrect equation, Y must equal £10.
It also indicates that equations (and the prices of) F, C, J, H, N, D, B and O must be correct.

As 3 out of 4 of the answers to the simultaneous equations investigated are X = 20, I assume that X must £20.
It also indicates that either equation (and the prices of) E or M is the incorrect one because when they are solved in a simultaneous equation, they produce a different answer to all the others.

Let X= £20 and Y= £10
I entered these values into all the equations to see if they fitted in with the figures.
All of them except E proved to be correct:
E. 18X + 12Y = 360 + 120 = 480; so the price for window E is incorrect.
This makes sense because E didn't produce the right answer when put in a simultaneous equation.

Well done to everyone who found the solution.
Can you see the similarity between the algebraic method and the 'comparing pairs' method?

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